reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem
  rng R c= dom S implies R"X c= (R*S)"(S.:X)
proof
  assume
A1: rng R c= dom S;
  let x be object;
  assume x in R"X;
  then consider Rx being object such that
A2: [x,Rx] in R and
A3: Rx in X by RELAT_1:def 14;
  Rx in rng R by A2,XTUPLE_0:def 13;
  then consider SRx being object such that
A4: [Rx,SRx] in S by A1,XTUPLE_0:def 12;
  SRx in S.:X & [x,SRx] in R*S by A2,A3,A4,RELAT_1:def 8,def 13;
  hence thesis by RELAT_1:def 14;
end;
